## Saturday, December 10, 2005

### Fine Structure for Markov Random Fields

A Markov Random Field is based on a set of variables that form the vertices of an adjacency graph. The Gibbs energy is decomposed in terms of the energies of configurations of maximal cliques of the adjacency graph.

We introduce a finer structure given by a simplicial complex whose 1-simplices are the edges of the adjacency graph. Every simplex is a clique of the adjacency graph. However, not all cliques need be simplices. We require that the Gibbs energy be decomposable in terms of the energies of configurations of simplices of the complex.

### Statistical Satisfiability

Using Markov Random Fields to explore issues of representation and reasoning.

#### SAT

Boolean Satisfiability (SAT) is the problem of finding satisfying valuations for a collection of Boolean constraints. Every set of Boolean constraints can be represented by a set of disjunctive propositional clauses.
k-SAT restricts attention to clauses with at most k literals. Every SAT problem can be reduced, by the addition of additional variables, to 3-SAT.

#### MRF

A Markov Random Field (MRF) determines a probability distribution over a set of random variables.
We view a MRF over a set of Boolean variables as a natural generalisation of SAT, and use a MRF to represent knowledge accumulated from experience.
We can construct a MRF, top-down a priori — for example, from a Markov Logic Network (MLN Richardson & Domingos), or bottom-up a posteriori, from a set of data.

Our bottom-up procedures are iterative and adaptive, so an initial MRF can be established to represent a priori assumptions, then adapted to reflect a posteriori experience.